Thank you. It was kind of short, because I didn't want it to be too detailed before I got my first flying-type pokemon. Here's the update:
- My character is again a female called Elesa.
- I picked Oshawott as my starter, defeated my rival, .
- I defeated the first gym and completed the Dream Mist quest.
- I caught my first flying-type pokemon! A Naughty Pidove called Swift (if I could, I'd name it Tailor Swift). It took me 10 Pidove's to get one with a good nature.
- I got to the girl with the stolen pokemon.
- When I walked into Wellspring Cave, I only needed to walk 4 steps (I think) to start my second battle against team Plasma, but after 2 steps, I encountered a wild Woobat. I caught it and he was a good nature as well (relaxed), and the best ability it can have (unaware), so I decided to keep it. I named it Confus.
- I trained them up quite a lot, but I still had problems while fighting Lenora, the 2nd (or 4th) gym leader.
- After Swift had turned itno a Tranquill, I was able to beat her.
- I helped her getting the Dragon Skull back trom Team Plasma.

I kinda missed rule 4 'cause I was reading through the rules so fast I must have missed it. I caught a Wurmple and used it against Brawly. That's the only time I broke that rule and I hope it doesn't mean I have to start again. I should have an update up later today beating up to and including Brawly.
Also, does general rule 8 (the one about HM slaves) still count for this challenge?

Eh, accidents happen. 'Specially when doin' a new Challenge for the first time. Just be sure t' not break any more rules.

Harmonia Rules' emphasis on constantly releasing and catching Pokemon and General Rules #8 intentionally clash with each other. You're allowed t' catch an HM slave whenever it's needed, ignoring any and all rules, but are still required to release 'em once you've obtained your new badge. This is 'cause General Rules #8 states you can catch HM slaves whenever needed. Doesn't say ya can ignore any Rules for anythin' but the catching of HM slaves.

... If anyone needs t' ask about a rule, and a long explanation like this is required in response, I feel like that rule needs to be fine tuned.

Haha, yeah, it is too bad. I leave tomorrow, so you won'y be seeing me for a week and a half or so. Meanwhile, I want to start a new challenge. Scramble/Monotype (Steel) with 6 pokemon. The reason I'm posting it now is because I want all the pokemon to be chosen when I come back.

I was quite lazy and consequently, I haven't gotten the update 'til over 24 hours after I said I would. But here it is:

HARMONIA CHALLENGE TEST UPDATE THINGY

Spoiler Alert!

I did all the starting stuff and off and picked Mudkip as my starter due to it having a type advantage against the first gym. Once I could catch more pokemon, I caught a Zigzagoon on my way to Petalburg. I [strike]voluntarily[strike] helped Wally catch his first pokemon and moved on to Rustboro. I didn't have 4 pokemon yet, so I caught a Nincada and Whismur in the route above Rustboro. The gym was incredibly easy and I only needed to use Mudkip.
Once the battle was done, I went to catch a Wurmple. I trained it up to level 12 with the help of all the trainers I avoided on the way to Rustboro. It evolved into Cascoon at level 7 and then again into Dustox at level 10, which I was really happy with. Gotta love how bug pokemon evolve so early . I then went to battle the Aqua Grunt in Rustboro Tunnel to make him give Peeko back to Mr. Briney. His Poochyena was no match for my Dustox and the battle was over in a few turns.
I then went to Dewford and delivered the letter to Steven without even using Flash (I've got my speedruns to thank for that. I can even navigate through Victory Road in the dark ). In the cave I caught a Makuhita, Abra and Zubat to complete my team of 4. I trained Zubat and Makuhita up to levels 15 and 13, respectively, and left Abra at level 10.
In the battle against Brawly, Dustox KOed Machop with 3 Confusions and got 2 off on Makuhita before fainting. I went into Zubat to try and finish it off. Supersonic hit first turn and Makuhita hit itself. Next turn I used Astonish and Makuhita hit itself again. Another Astonish took out Makuhita and got me the bagde.

Feedback
I think that having to catch 4 pokemon for the beginning couple of gyms is overkill. Maybe there could be a varying number of pokemon needed depending on where you are in the game. For the first gym I only used 1 and for the second gym I only used 2 pokemon.
However, I think this is quite an enjoyable challenge from what I've experienced so far.

I agree. Four is definitely too large of a (forced) team early on in the game... But I'm not sure how t' handle that issue. Lowering it to three was my first consideration. Your suggestion I also considered, though I dunno how t' time the team size changes. Perhaps team size should start at two, and increases by one every two badges?